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Full text of "Analytical Mechanics"

142
ANALYTICAL MECHANICS
115. Center of Mass of Continuous Bodies.—When the particles form a continuous body we can replace the summation signs of equation (I') by integration signs and obtain the following expressions for the coordinates of the center of mass:
x=^
r' dm .
y dm
y—r—>
I   dm
Jo
r,
Jo
z dm
r
Jo
dm
(I)
where m is the mass of the body.
ILLUSTRATIVE EXAMPLES.
1. Find the center of mass of the parabolic lamina bounded by the curves ?/ = 2 px and x = a, Fig. 74.
Obviously the center of mass lies on the re-axis.   Therefore we need to
0                   4a
'----^
FIG. 74.                                             FIG. 75.
* In general if y is a function of x then the average value of y between the limits Xi and x* is given by the relation: y = —--— f^ydx.